## Overview

In January 2015, SIAM published my book titled Model Emergent Dynamics in Complex Systems on rational mathematical modelling of dynamics.

Via the web page https://tuck.adelaide.edu.au/modelling.php you may create a PDF book of draft supplementary material tailored to the interests you specify.

Maurício Kritz (2019) commented in reference to the book: develops methods, techniques, and algorithms for studying the behaviour of systems ... necessity of having simple models, equations and mathematics as brilliantly explained ... the huge and beautiful knowledge accumulated recently about dynamical systems.

## Errata of printed first edition

• p.8, Fig.1.1: $$x=1-\frac12\epsilon+\cdots$$, not $$+\frac12\epsilon$$.
• p.52, l.-6: $$\lim_{x\to0}\frac{\cos x}2$$ not the negative.
• p.60, Ex. 2.11: the ODE $$x^2y''+xy'+xy^2=0$$, not $$x^2y''+x^2y'+xy^3=0$$.
• p.61, Ex. 2.14: change the hint to writing $$x^2y''-xy'+y$$ as $$x^2[x(y/x)']'$$.
• p.75, l.7: $$\vartheta$$ denotes the phase of the left-travelling wave (not the right-travelling).
• p.158, Ex. 4.6 (a slow subspace): use $$\dot x=-ay$$, not $$\dot x=-ax$$.
• p.160, Ex. 4.12 (instability at higher order): construct the slow manifold to higher order, such as $$O(x^3+y^3)$$, and find that the conclusion is not quite correct!
• p.206, Ex 5.4 and 5.5: in the answers there is no need for the error component $$,\alpha^2$$'', omit.
• p.246, Eqn (7.5): omit the $$\theta$$ derivatives (in the last term).
• p.265--6, Eqn (7.20): the right-hand side should be "$$=-k(\cdots)$$", not "$$=k(\cdots)$$"; consequently, the second iteration answer should be "$$\partial s/\partial t \approx (k/s)\left[1-\frac13k+\frac13s_x^2+\frac13ss_{xx} -\frac23ks_x^2\right]$$".
• p.253: last line of the first displayed maths should read $$=(uU-U^2)w\frac{\partial^2C}{\partial x^2}$$.
• p.299, just before section 9.1.4: "$$[-2\gamma +\frac23 gamma^2+$$" should be "$$[-2\gamma -\frac23gamma^2+$$".
• p.300: "decay rate $$\pi/4$$" should be "decay rate $$\pi^2/4$$".
• p.304, lateral momentum equation: the RHS of the update PDE needs a factor of $$1/Re$$; the residual of the tangential stress equation needs to be negated; and "change to the pressure field is due to viscous stresses at the free surface and the need to accelerate the flow vertically" should read "change to the horizontal velocity field is due to viscous drag at the bed, flagged by factor~$$\gamma$$, and the acceleration due to spatial variations".
• p.305, last line before Iterative construction'': boundary condition "$$\hat u=0$$" should read "$$\partial\hat u/\partial Z=0$$".
• p.474, line 13, capital $$U$$ in the PDE should be little $$u$$.
• p.490, in the first paragraph the $$f_{2j}$$ should be in the numerators of the fractions, not the denominators. Similarly in the second paragraph, $$f_{2,\min}$$ should also be in the numerator.
• p.708--712, Algorithm 21.4--8 does not work when there is large noise in the slow variables. It does appear to be OK when the slow variable noise is small or zero.

## Among the chapters that follow, choose the interesting ones

The chapters labelled BOOK are in the published book. At the end click Submit to process, typeset and produce a PDF supplement tailored just for you. The processing automatically detects all prerequisite chapters and includes them as well. Only select what interests you.

But first, what computer algebra package do you prefer?
Reduce (recommended)
Maple (as yet only available in Part I)
Mathematica (as yet only available in Part I, thanks to Rick Eller)

Asymptotic methods solve algebraic and differential equations
BOOK Perturbed algebraic equations solved iteratively
BOOK Power series solve ordinary differential equations
BOOK The normal form of oscillations give their amplitude and frequency

Basic fluid dynamics
Flow description
Conservation of mass
Conservation of momentum
The state space

Centre manifolds emerge
Couette flow
The metastability of quasi-stationary probability distributions
BOOK The centre manifold emerges
BOOK Construct centre manifolds iteratively
Taylor vortices form in a pitchfork bifurcation
Irregular centre manifolds encompass novel applications

High fidelity numerical models use centre manifolds
Introduction to some numerical methods
BOOK Introduce holistic discretisation on just two elements
BOOK Holistic discretisation in one-D
Model physical boundary conditions
Dispersion along pipes invokes cylindrical elements

Normal forms usefully illustrate
BOOK Normal forms simplify evolution
BOOK Separate the fast and slow dynamics
BOOK Appropriate initial conditions empower accurate forecasts
BOOK Subcentre slow manifolds are useful but do not emerge
N/A Normal forms for homogenisation

Hopf bifurcation has oscillation within the centre manifold
Linear stability of double diffusion
BOOK Directly model oscillations in Cartesian-like variables
BOOK Model the modulation of oscillations
Nonlinear evolution of double diffusion
N/A Chaos appears in triple convection

Avoid memory in modelling non-autonomous or stochastic systems
BOOK Averaging is often a good first modelling approximation
BOOK Coordinate transforms separate slow from fast in non-autonomous dynamics
BOOK Introducing basic stochastic calculus
BOOK Strong and weak models of stochastic dynamics

Large-scale spatial variations
Poiseuille pipe flow
BOOK Cross-stream mixing causes longitudinal dispersion in pipes
BOOK Thin fluid films evolve slowly over space and time
BOOK Resolve inertia in thicker faster fluid films

Patterns form and evolve
One-dimensional introduction
N/A Two-dimensional spatio-temporal patterns
N/A Embedding slow dynamics

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